Basically it allows actions that usually would be performed by a page refresh through PHP to get more information from a database to be performed a page refresh such examples can be seen if you go to scriptalicious looks like a Ajax API. I have never used Flash so I can't speak of its replacability.

Flash is completely different than AJAX and believe me if you aren't a designer then don't go for Flash (Actionscript). Ajax give you power of server side coding staying on same page, mean you don't have to refresh page (Contact server) to fulfill a request. Using Javascript and XML calls you can perform the same thing which will be displayed after page refresh. Many people are describing AJAX as a tool which helps in solving the problem of page refresh but its not true.